Smoothing low-dimensional algebraic cycles [after Koll\'ar and Voisin]

Abstract

Let $X$ be a smooth projective complex algebraic variety. An old question of Borel and Haefliger asks whether any (possibly singular) algebraic subvariety of $X$ is homologically equivalent to a linear combination with integral coefficients of smooth algebraic subvarieties of $X$. In general, this question is too optimistic, and counterexamples have been known for a long time. The aim of this survey is to explain how J\'anos Koll\'ar and Claire Voisin have provided a positive answer to Borel and Haefliger's question, for subvarieties of dimension less than half the dimension of $X$.